Artículos con la etiqueta ‘Lógica y ciencias de la computación’

We prove that the expressive power of first-order logic with team semantics plus contradictory negation does not rise beyond that of first-order logic (with respect to sentences), and that the totality atoms of arity k +1 are not definable in terms of the totality atoms of arity k. We furthermore prove that all first-order nullary and unary dependencies are strongly first order, in the sense that they do not increase the expressive power of first order logic if added to it.

We seize the opportunity of the publication of selected papers from the \emph{Logic, categories, semantics} workshop in the \emph{Journal of Applied Logic} to survey some current trends in logic, namely intuitionistic and linear type theories, that interweave categorical, geometrical and computational considerations. We thereafter present how these rich logical frameworks can model the way language conveys meaning.

In work of Fokkinga and Meertens a calculational approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting retain the vital type information, but poorly express the reasoning and development of categorical proofs. In order to combine the strengths of these two perspectives, we propose the use of string diagrams, common folklore in the category theory community, allowing us to retain the type information whilst pursuing a calculational form of proof. These graphical representations provide a topological perspective on categorical proofs, and silently handle functoriality and naturality conditions that require awkward bookkeeping in more traditional notation.

We prove an extensionality theorem for the “type-in-type” dependent type theory with Sigma-types. We suggest that the extensional equality type be identified with the logical equivalence relation on the free term model of type theory.

In recent years, a number of researchers have suggested that the extensional equality in type theory is the canonical logical relation defined by induction on type structure (Tait [1998], Altenkirch et al [2008], Coquand [2010], Harper et al [2013]). Here we make this position explicit in the statement of Extensionality Thesis.

Simple and useful classical logic is unfortunately defective with its problematic definition of material implication. This paper presents an implication relation defined by a simple equation to replace the traditional material implication in classical logic. Common “paradoxes” of material implication are avoided while simplicity and usefulness of the system are reserved with this implication relation.

Recursive maps, nowadays called primitive recursive maps, PR maps, have been introduced by G\”odel in his 1931 article for the arithmetisation, g\”odelisation, of metamathematics. For construction of his undecidable formula he introduces a non-constructive, non-recursive predicate beweisbar, provable. Staying within the area of categorical free-variables theory PR of primitive recursion or appropriate extensions opens the chance to avoid the two (original) G\”odel’s incompleteness theorems: these are stated for Principia Mathematica und verwandte Systeme, “related systems” such as in particular Zermelo-Fraenkel set theory ZF and v. Neumann G\”odel Bernays set theory NGB. On the basis of primitive recursion we consider \mu-recursive maps as partial pr maps. Special terminating general recursive maps considered are complexity controlled iterations. Map code evaluation is then given in terms of such an iteration. We discuss iterative pr map code evaluation versus termination conditioned soundness and based on this decidability of primitive recursive predicates.

Based on graphic lambda calculus, we propose a program for a new model of asynchronous distributed computing, inspired from Hewitt Actor Model, as well as several investigation paths, concerning how one may graft lambda calculus and knot diagrammatics.

As one of the longest-running computer-assisted formal mathematics projects, large tracts of mathematical knowledge have been formalized with the help of the Mizar system. Because Mizar is based on first-order classical logic and set theory, and because of its emphasis on pure mathematics, the Mizar library offers a cornucopia for the researcher interested in foundations of mathematics. With Mizar, one can adopt an experimental approach and take on problems in foundations, at least those which are amenable to such experimentation. Addressing a question posed by H. Friedman, we use Mizar to take on the question of surveying the sentence complexity (measured by quantifier alternation) of mathematical theorems. We find, as Friedman suggests, that the sentence complexity of most Mizar theorems is universal (Π 1 , or ∀ ), and as one goes higher in the sentence complexity hierarchy the number of Mizar theorems having these complexities decreases rapidly. The results support the intuitive idea that mathematical statements, even when carried out an abstract set-theoretical style, are usually quite low in the sentence complexity hierarchy (not more complex than ∀∃∀ or ∃∀ ).

In addition to their limpid interface with semantics, categorial grammars enjoy another important property: learnability. This was first noticed by Buskowsky and Penn and further studied by Kanazawa, for Bar-Hillel categorial grammars. What about Lambek categorial grammars? In a previous paper we showed that product free Lambek grammars where learnable from structured sentences, the structures being incomplete natural deductions. These grammars were shown to be unlearnable from strings by Foret and Le Nir. In the present paper we show that Lambek grammars, possibly with product, are learnable from proof frames that are incomplete proof nets. After a short reminder on grammatical inference \`a la Gold, we provide an algorithm that learns Lambek grammars with product from proof frames and we prove its convergence. We do so for 1-valued also known as rigid Lambek grammars with product, since standard techniques can extend our result to $k$-valued grammars. Because of the correspondence between cut-free proof nets and normal natural deductions, our initial result on product free Lambek grammars can be recovered.